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arXiv · 2103.13075

Uniqueness and continuity of the solution to $L_p$ dual Minkowski problem

Hejun Wang·Jiazu Zhou
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Abstract

Lutwak, Yang and Zhang \cite{LYZ2018} introduced the $L_p$ dual curvature measure that unifies several other geometric measures in dual Brunn-Minkowski theory and Brunn- Minkowski theory. Motivated by works in \cite{LYZ2018}, we consider the uniqueness and continuity of the solution to the $L_p$ dual Minkowski problem. To extend the important work (Theorem \ref{uniquepolytope}) of LYZ to the case for general convex bodies, we establish some new Minkowski-type inequalities which are closely related to the optimization problem associated with the $L_p$ dual Minkowski problem. When $q< p$, the uniqueness of the solution to the $L_p$ dual Minkowski problem for general convex bodies is obtained. Moreover, we obtain the continuity of the solution to the $L_p$ dual Minkowski problem for convex bodies.

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Hejun Wang, Jiazu Zhou. 2021-03-24. Uniqueness and continuity of the solution to $L_p$ dual Minkowski problem. https://arxiv.org/abs/2103.13075

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